Trigonometric equation cos sin and power

Question

The problem is $2\cos t - 3\sin^2t +2 = 0$. I get to $2\cos t -3\sin^2t =-2$ I think that I need to use a trigonometric identity like $\cos(x+y)$ and to divide $2\cos t -3\sin^2t$ with the $\sqrt{2^2+3^2}$

Do you know how to solve this? It should be $\sqrt{2^2 + 3^2}$

Answer 1

$$2 \cos t - 3 \sin^2t +2 = 0\\.$$ Write $$\sin^2 t=1-\cos^2 t, $$ then you have a quadratic equation. Solve for $\cos t$ $$ 2\cos t -3(1-\cos^2 t)+2=0\\3\cos ^2 t+2\cos t-3+2=0\\\cos t=-1,\;\frac{1}{3}. $$

Answer 2

We have, $$2\cos t-3\sin^2 t+2=0$$ $$\implies 2\cos t-3+3\cos^2 t+2=0$$ $$\implies 3\cos^2t+ 2\cos t-1=0$$ Factorizing the expression, we get $$(3\cos t-1)(\cos t+1)=0$$ $$\text{if}\ 3\cos t-1=0 \implies \cos t=\frac{1}{3}\implies \color{blue}{t=2n\pi\pm\cos^{-1}\left(\frac{1}{3}\right)}$$ $$\text{if}\ \cos t+1=0 \implies \cos t=-1 \implies \color{blue}{t=(2n+1)\pi}$$ Where, $n$ is any integer